InTech-Neural Control Toward a Unified Intelligent Control Design Framework for Nonlinear Systems

7/27/2019 InTech-Neural Control Toward a Unified Intelligent Control Design Framework for Nonlinear Systems

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5

Neural Control Toward a Unified IntelligentControl Design Framework for

Nonlinear Systems

Dingguo Chen1, Lu Wang2, Jiaben Yang3 and Ronald R. Mohler41Siemens Energy Inc., Minnetonka, MN 55305

2Siemens Energy Inc., Houston, TX 770793Tsinghua University, Beijing 100084

4

Oregon State University, OR 973301,2,4USA3China

1. Introduction

There have been significant progresses reported in nonlinear adaptive control in the last twodecades or so, partially because of the introduction of neural networks (Polycarpou, 1996;Chen & Liu, 1994; Lewis, Yesidirek & Liu, 1995; Sanner & Slotine, 1992; Levin & Narendra,1993; Chen & Yang, 2005). The adaptive control schemes reported intend to design adaptiveneural controllers so that the designed controllers can help achieve the stability of theresulting systems in case of uncertainties and/or unmodeled system dynamics. It is a typicalassumption that no restriction is imposed on the magnitude of the control signal.Accompanied with the adaptive control design is usually a reference model which isassumed to exist, and a parameter estimator. The parameters can be estimated within a pre-designated bound with appropriate parameter projection. It is noteworthy that these designapproaches are not applicable for many practical systems where there is a restriction on thecontrol magnitude, or a reference model is not available.On the other hand, the economics performance index is another important objective forcontroller design for many practical control systems. Typical performance indexes include,for instance, minimum time and minimum fuel. The optimal control theory developed a few

decades ago is applicable to those systems when the system model in question along with aperformance index is available and no uncertainties are involved. It is obvious that theseoptimal control design approaches are not applicable for many practical systems wherethese systems contain uncertain elements.Motivated by the fact that many practical systems are concerned with both system stabilityand system economics, and encouraged by the promising images presented by theoreticaladvances in neural networks (Haykin, 2001; Hopfield & Tank, 1985) and numerous applicationresults (Nagata, Sekiguchi & Asakawa, 1990; Methaprayoon, Lee, Rasmiddatta, Liao & Ross,2007; Pandit, Srivastava & Sharma, 2003; Zhou, Chellappa, Vaid & Jenkins, 1998; Chen & York,2008; Irwin, Warwick & Hunt, 1995; Kawato, Uno & Suzuki, 1988; Liang 1999; Chen & Mohler,1997; Chen & Mohler, 2003; Chen, Mohler & Chen, 1999), this chapter aims at developing an

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Recent Advances in Robust Control Novel Approaches and Design Methods92

intelligent control design framework to guide the controller design for uncertain, nonlinearsystems to address the combining challenge arising from the following:

The designed controller is expected to stabilize the system in the presence ofuncertainties in the parameters of the nonlinear systems in question.

The designed controller is expected to stabilize the system in the presence ofunmodeled system dynamics uncertainties.

The designed controller is confined on the magnitude of the control signals. The designed controller is expected to achieve the desired control target with minimum

total control effort or minimum time.The salient features of the proposed control design framework include: (a) achieving nearlyoptimal control regardless of parameter uncertainties; (b) no need for a parameter estimatorwhich is popular in many adaptive control designs; (c) respecting the pre-designated rangefor the admissible control.Several important technical aspects of the proposed intelligent control design frameworkwill be studied:

Hierarchical neural networks (Kawato, Uno & Suzuki, 1988; Zakrzewski, Mohler &Kolodziej, 1994; Chen, 1998; Chen & Mohler, 2000; Chen, Mohler & Chen, 2000; Chen,Yang & Moher, 2008; Chen, Yang & Mohler, 2006) are utilized; and the role of each tierof the hierarchy will be discussed and how each tier of the hierarchical neural networksis constructed will be highlighted.

The theoretical aspects of using hierarchical neural networks to approximately achieveoptimal, adaptive control of nonlinear, time-varying systems will be studied.

How the tessellation of the parameter space affects the resulting hierarchical neuralnetworks will be discussed.

In summary, this chapter attempts to provide a deep understanding of what hierarchical

neural networks do to optimize a desired control performance index when controllinguncertain nonlinear systems with time-varying properties; make an insightful investigationof how hierarchical neural networks may be designed to achieve the desired level of controlperformance; and create an intelligent control design framework that provides guidance foranalyzing and studying the behaviors of the systems in question, and designing hierarchicalneural networks that work in a coordinated manner to optimally, adaptively control thesystems.This chapter is organized as follows: Section 2 describes several classes of uncertainnonlinear systems of interest and mathematical formulations of these problems arepresented. Some conventional assumptions are made to facilitate the analysis of theproblems and the development of the design procedures generic for a large class of

nonlinear uncertain systems. The time optimal control problem and the fuel optimal controlproblem are analyzed and an iterative numerical solution process is presented in Section 3.These are important elements in building a solution approach to address the controlproblems studied in this paper which are in turn decomposed into a series of controlproblems that do not exhibit parameter uncertainties. This decomposition is vital in theproposal of the hierarchical neural network based control design. The details of thehierarchical neural control design methodology are given in Section 4. The synthesis ofhierarchical neural controllers is to achieve (a) near optimal control (which can be time-optimal or fuel-optimal) of the studied systems with constrained control; (b) adaptivecontrol of the studied control systems with unknown parameters; (c) robust control of thestudied control systems with the time-varying parameters. In Section 5, theoretical results

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Neural Control Toward a UnifiedIntelligent Control Design Framework for Nonlinear Systems 93

are developed to justify the fuel-optimal control oriented neural control design proceduresfor the time-varying nonlinear systems. Finally, some concluding remarks are made.

2. Problem formulation

As is known, the adaptive control design of nonlinear dynamic systems is still carried out on aper case-by-case basis, even though there have numerous progresses in the adaptive of lineardynamic systems. Even with linear systems, the conventional adaptive control schemes havecommon drawbacks that include (a) the control usually does not consider the physical controllimitations, and (b) a performance index is difficult to incorporate. This has made the adaptivecontrol design for nonlinear system even more challenging. With this common understanding,this Chapter is intended to address the adaptive control design for a class of nonlinear systemsusing the neural network based techniques. The systems of interest are linear in both controland parameters, and feature time-varying, parametric uncertainties, confined control inputs,and multiple control inputs. These systems are represented by a finite dimensional differential

system linear in control and linear in parameters.The adaptive control design framework features the following:

The adaptive, robust control is achieved by hierarchical neural networks. The physical control limitations, one of the difficulties that conventional adaptive

control can not handle, are reflected in the admissible control set.

The performance measures to be incorporated in the adaptive control design, deemedas a technical challenge for the conventional adaptive control schemes, that will beconsidered in this Chapter include:

Minimum time resulting in the so-called time-optimal control Minimum fuel resulting in the so-called fuel-optimal control

Quadratic performance index resulting in the quadratic performance optimalcontrol.Although the control performance indices are different for the above mentioned approaches,the system characterization and some key assumptions are common.The system is mathematically represented by

( ) ( ) ( )x a x C x p B x u= + +$ (1)

where nx G R is the state vector, lpp R is the bounded parameter vector,mu R

is the control vector, which is confined to an admissible control set U,

[ ]1 2( ) ( ) ( ) ( )na x a x a x a x

= A is an n -dimensional vector function of x ,

11 12 1

21 22 2

1 2

( ) ( ) ...

( ) ( ) ... ( )( )

... ... ... ...

( ) ( ) ... ( )

l

l

n n nl

C x C x C

C x C x C xC x

C x C x C x

=

is an n l -dimensional matrix function of x , and

11 12 1

21 22 2

1 2

( ) ( ) ...

( ) ( ) ... ( )( )

... ... ... ...

( ) ( ) ... ( )

m

m

n n nm

B x B x B

B x B x B xB x

B x B x B x

=

is an n m -dimensional matrix function of x .

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The control objective is to follow a theoretically sound control design methodology to

design the controller such that the system is adaptively controlled with respect to

parametric uncertainties and yet minimizing a desired control performance.To facilitate the theoretical derivations, several conventional assumptions are made in the

following and applied throughout the Chapter.AS1: It is assumed that (.)a , (.)C and (.)B have continuous partial derivatives with respect

to the state variables on the region of interest. In other words, ( )ia x , ( )isC x , ( )ikB x ,( )i

j

a x

x

,

( )is

j

C x

x

, and

( )ik

j

B x

x

for , 1,2, ,i j n= A ; 1,2, ,k m= A ; 1,2, ,s l= A exist and are continuous

and bounded on the region of interest.

It should be noted that the above conditions imply that (.)a , (.)C and (.)B satisfy the

Lipschitz condition which in turn implies that there always exists a unique and continuous

solution to the differential equation given an initial condition 0 0( )x t = and a bounded

control ( )u t .

AS2: In practical applications, control effort is usually confined due to the limitation of

design or conditions corresponding to physical constraints. Without loss of generality,

assume that the admissible control set U is characterized by:

{ }:| | 1, 1,2, ,iU u u i m= = A (2)

where iu is u 's i th component.

AS3: It is assumed that the system is controllable.

AS4: Some control performance criteria J may relate to the initial time 0t and the final time

t . The cost functional reflects the requirement of a particular type of optimal control.

AS5: The target set is defined as { }: ( ( )) 0f fx x t = = where i s ( 1,2, ,i q= A ) are thecomponents of the continuously differentiable function vector (.) .

Remark 1: As a step of our approach to address the control design for the system (1), the

above same control problem is studied with the only difference that the parameters in Eq.

(1) are given. An optimal solution is sought to the following control problem:The optimal control problem ( 0P ) consists of the system equation (1) with fixed and known

parameter vector p , the initial time 0t , the variable final time ft , the initial state 0 0( )x x t= ,together with the assumptions AS1, AS2, AS3, AS4, AS5 satisfied such that the system state

conducts to a pre-specified terminal set f at the final time ft while the control

performance index is minimized.

AS6: There do not exist singular solutions to the optimal control problem ( 0P ) as described

in Remark 1 (referenced as the control problem ( 0P ) later on distinct from the original

control problem ( P )).

AS7:x

p

is bounded on pp and xx .

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Remark 2: For any continuous function ( )x defined on the compact domain nx R ,

there exists a neural network characterized by ( )fNN x such that for any positive number

*f ,

*| ( ) ( )|f ff x NN x < .

AS8: Let the sufficiently trained neural network be denoted by ( , )sNN x , and the neural

network with the ideal weights and biases by *( , )NN x where s and * designate the

parameter vectors comprising weights and biases of the corresponding neural networks.

The approximation of ( , )f sNN x to *( , )fNN x is measured by

* *( ; ; ) | ( , ) ( , )|f s f s fNN x NN x NN x = . Assume that *( ; ; )f sNN x is bounded by a

pre-designated number 0,s > i.e., *( ; ; )s

f sNN x < .

AS9: The total number of switch times for all control components for the studied fuel-optimal control problem is greater than the number of state variables.

Remark 3: AS9 is true for practical systems to the best knowledge of the authors. Theassumption is made for the convenience of the rigor of the theoretical results developed inthis Chapter.

2.1 Time-optimal control

For the time-optimal control problem, the system characterization, the control objective,constraints remain the same as for the generic control problem with the exception that thecontrol performance index reflected in the Assumption AS4 is replaced with the following:

AS4: The control performance criteria is

0

1ft

t

J ds= where 0t and t are the initial time and the

final time, respectively. The cost functional reflects the requirement of time-optimal control.

2.2 Fuel-optimal control

For the fuel-optimal control problem, the system characterization, the control objective,constraints remain the same as for the time-optimal control problem with the AssumptionAS4 replaced with the following:

AS4: The control performance criteria is

0

0 1| |

ftm

k kkt

J e e u ds=

= + where 0t and t are the

initial time and the final time, respectively, and ke ( 0,1,2, ,k m= A ) are non-negative

constants. The cost functional reflects the requirement of fuel-optimal control as related tothe integration of the absolute control effort of each control variable over time.

2.3 Optimal control with quadratic performance indexFor the quadratic performance index based optimal control problem, the systemcharacterization, the control objective, constraints remain the same with the AssumptionAS4 replaced with the following:AS4: The control performance criteria is

0

1 1( ( ) ( )) ( )( ( ) ( )) ( ) ( )

2 2

ft

f f f f f e e

t

J x t r t S t x t r t x Qx u u R u u ds = + + where 0t and ft are

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the initial time and the final time, respectively; and ( ) 0fS t , 0Q , and 0R with

appropriate dimensions; and the desired final state ( )fr t is the specified as the equilibrium

ex , and eu is the equilibrium control.

3. Numerical solution schemes to the optimal control problems

To solve for the optimal control, mathematical derivations are presented below for each ofthe above optimal control problems to show that the resulting equations represent theHamiltonian system which is usually a coupled two-point boundary-value problem(TPBVP), and the analytic solution is not available, to our best knowledge. It is worth notingthat in the solution process, the parameter is assumed to be fixed.

3.1 Numerical solution scheme to the time optimal control problem

By assumption AS4, the optimal control performance index can be expressed as

00( ) 1

ft

tJ t dt=

where 0t is the initial time, and ft is the final time.

Define the Hamiltonian function as

( , , ) 1 ( ( ) ( ) ( ) )H x u t a x C x p B x u= + + +

where [ ]1 2 n

= A is the costate vector.

The final-state constraint is ( ( )) 0fx t = as mentioned before.The state equation can be expressed as

0( ) ( ) ( ) ,H

x a x C x p B x u t t

= = + +

$

The costate equation can be written as

( ( ) ( ) ( ) ),

a x C x p B x uHt T

x x

+ +

= =

$

The Pontryagin minimum principle is applied in order to derive the optimal control (Lee &Markus, 1967). That is,

* * * * *( , , , ) ( , , , )H x u t H x u t

for all admissible u .

where *u , *x and * correspond to the optimal solution.

Consequently,

* * *

1 1( ) ( )

m mk k k kk k

B x u B x u = =

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where ( )kB x is the k th column of the ( )B x .

Since the control components ku 's are all independent, the minimization of 1 ( )m

k kkB x u

=

is equivalent to the minimization of ( )k kB x u

.

The optimal control can be expressed as * *sgn( ( ))k ku s t= , where sgn(.) is the sign function

defined as sgn( ) 1t = if 0t > or sgn( ) 1t = if 0t < ; and ( ) ( )k ks t B x= is the k th

component of the switch vector ( ) ( )S t B x = .

It is observed that the resulting Hamiltonian system is a coupled two-point boundary-valueproblem, and its analytic solution is not available in general.With assumption AS6 satisfied, it is observed from the derivation of the optimal time control

that the control problem ( 0P ) has bang-bang control solutions.

Consider the following cost functional:

0

2

1

1 ( ( ))f

qt

i i fti

J dt x t =

= +

where i 's are positive constants, and i 's are the components of the defining equation of

the target set { }: ( ( )) 0f fx x t = = to the system state is transferred from a given initial stateby means of proper control, and q is the number of components in .

It is observed that the system described by Eq. (1) is a nonlinear system but linear in control.With assumption AS6, the requirements for applying the Switching-Time-Varying-Method(STVM) are met. The optimal switching-time vector can be obtained by using a gradient-based method. The convergence of the STVM is guaranteed if there are no singularsolutions.Note that the cost functional can be rewritten as follows:

0

' '0 0[( ( ) ( ), )]

ft

tJ a x b x u dt= + < >

where '0 1( ) 1 2 , ( ) ( ) ,q i

i iia x a x C x p

x

=

= + < + >

'0 1( ) 2 [ ] ( )

q ii ii

b x B xx

=

=

, and ( )a x ,

( )C x , p and ( )B x are as given in the control problem ( 0P ).

Define a new state variable 0( )x t as follows:

' '0 0 00( ) [( ( ) ( ), )]

t

tx t a x b x u dt= + < >

Define the augmented state vector 0x x x

= ,'0( ) ( ) ( ( ) ( ) )a x a x a x C x p

= + , and

'0( ) ( ) ( ( ))B x b x B x

= .

The system equation can be rewritten in terms of the augmented state vector as

( ) ( )x a x B x u= +$ where 0 0( ) 0 ( )x t x t

= .

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A Hamiltonian system can be constructed for the above state equation with the costateequation given by

( ( ) ( ) )a x B x ux

= +

$ where ( ) | ( )f f

Jt x t

x

=

.

It has been shown (Moon, 1969; Mohler, 1973; Mohler, 1991) that the number of the optimalswitching times must be finite provided that no singular solutions exist. Let the zeros of

( )ks t be ,k j+ ( 1,2, ,2 kj N

+= A , 1,2, ,k m= A ; and1 2, ,k j k j

+ +< for 1 21 2 kj j N+ < ).

*,2 1 ,2

1

( ) [sgn( ) sgn( )].kN

k k j k jj

u t t t

+

+ +

=

=

Let the switch vector for the k th component of the control vector be k kN N +

= where

,1 ,2k

k

Nk k N

+

++ + =

A . Let 2k kN N

+= . Then kN is the switching vector of kN

dimensions.

Let the vector of switch functions for the control variable ku be defined as

1 2k k k

k

N N N

N

+ =

A where 1 ,( 1) ( )kN j

k k jj s += ( 1,2, ,2 kj N

+= A ).

The gradient that can be used to update the switching vector kN can be given by

k

Nk

NJ

=

The optimal switching vector can be obtained iteratively by using a gradient-based method.

, 1 , ,k k kN i N i N k iK + = +

where ,k iK is a properly chosen k kN N -dimensional diagonal matrix with non-negative

entries for the i th iteration of the iterative optimization process; and ,kN i represents the

i th iteration of the switching vector kN .

Remark 4: The choice of the step sizes as characterized in the matrix ,k iK must consider two

facts: if the step size is chosen too small, the solution may converge very slowly; if the stepsize is chosen too large, the solution may not converge. Instead of using the gradientdescent method, which is relatively slow compared to other alternative such as methodsbased on Newton's method and inversion of the Hessian using conjugate gradienttechniques.

When the optimal switching vectors are determined upon convergence, the optimal control

trajectories and the optimal state trajectories are computed. This process will be repeated forall selected nominal cases until all needed off-line optimal control and state trajectories are

obtained. These trajectories will be used in training the time-optimal control oriented neural

networks.

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3.2 Numerical solution scheme to the fuel optimal control problem

By assumption AS4, the optimal control performance index can be expressed as

00 0 1

( ) | |f

mt

k kt k

J t e e u dt=

= +

where 0t is the initial time, and ft is the final time.

Define the Hamiltonian function as

01

( , , ) | | ( ( ) ( ) ( ) )m

k kk

H x u t e e u a x C x p B x u=

= + + + +

where [ ]1 2 n

= A is the costate vector.

The final-state constraint is ( ( )) 0fx t = as mentioned before.

The state equation can be expressed as

0( ) ( ) ( ) ,H

x a x C x p B x u t t

= = + +

$

The costate equation can be written as

0 1

( ( ) ( ) ( ) )

( | |) ( ( ) ( ) ( ) ),

mk kk

a x C x p B x uH

x x

e e u a x C x p B x ut T

x x

=

+ + = = +

+ + +=

The Pontryagin minimum principle is applied in order to derive the optimal control (Lee &Markus, 1967). That is,

* * * * *( , , , ) ( , , , )H x u t H x u t for all admissible u , where*u , *x and * correspond to the

optimal solution.Consequently,

* * * *

1 1

1 1

| | ( )

| | ( )

m mk k k kk k

m mk k k kk k

e u B x u

e u B x u

= =

= =

+

+

where ( )kB x is the k th column of the ( )B x .

Since the control components ku 's are all independent, the minimization of

1 1| | ( )

m mk k k kk k

e u B x u= =

+ is equivalent to the minimization of | | ( )k k k ke u B x u

+ .

Since 0ke , define ( ) /k k ks B x e= . The fuel-optimal control satisfies the following

condition:* *

* *

*

sgn( ( )),| ( )| 1

0,| ( )| 1

,| ( )| 1

k k

k k

k

s t s t

u s t

undefined s t

>

=


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where 1,2, ,k m= A .

Note that the above optimal control can be written in a different form as follows:

* * *k k ku u u

+ = +

where * *1

sgn( ( ) 1) 12

k ku s t+ = + , and

* *1 sgn( ( ) 1) 12

k ku s t = + .

It is observed that the resulting Hamiltonian system is a coupled two-point boundary-valueproblem, and its analytic solution is not available in general.With assumption AS6 satisfied, it is observed from the derivation of the optimal fuel control

that the control problem ( 0P ) only has bang-off-bang control solutions.

Consider the following cost functional:

0

20

1 1

| | ( ( ))f

qmt

k k i i f t k i

J e e u dt x t = =

= + +

where i 's are positive constants, and i 's are the components of the defining equation of

the target set { }: ( ( )) 0f fx x t = = to the system state is transferred from a given initial stateby means of proper control, and q is the number of components in .

It is observed that the system described by Eq. (1) is a nonlinear system but linear in control.With assumption AS6, the requirements for the STVM's application are met. The optimalswitching-time vector can be obtained by using a gradient-based method. The convergenceof the STVM is guaranteed if there are no singular solutions.Note that the cost functional can be rewritten as follows:

0

' '0 0

1

[( ( ) ( ), ) | |]f

mt

k ktk

J a x b x u e u dt=

= + < > +

where '0 0 1( ) 2 , ( ) ( ) ,q i

i iia x e a x C x p

x

=

= + < + >

'0 1( ) 2 [ ] ( )

q ii ii

b x B xx

=

=

, and ( )a x ,

( )C x , p and ( )B x are as given in the control problem ( 0P ).

Define a new state variable 0( )x t as follows:

' '

0 0 00 1

( ) [( ( ) ( ), ) | |]mt

k kt k

x t a x b x u e u dt=

= + < > +

Define the augmented state vector 0x x x

= ,

'0( ) ( ) ( ( ) ( ) )a x a x a x C x p

= + , and

'0( ) ( ) ( ( ))B x b x B x

= .

The system equation can be rewritten in terms of the augmented state vector as

( ) ( )x a x B x u= +$ where 0 0( ) 0 ( )x t x t

= .

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The adjoint state equation can be written as

( ( ) ( ) )a x B x ux

= +

$ where ( ) | ( )fJ

t x tx

=

.

It has been shown (Moon, 1969; Mohler, 1973; Mohler, 1991) that the number of the optimalswitching times must be finite provided that no singular solutions exist. Let the zeros of

( ) 1ks t be ,k j+ ( 1,2, ,2 kj N

+= A , 1,2, ,k m= A ; and1 2, ,k j k j

+ +< for 1 21 2 kj j N+ < ) which

represent the switching times corresponding to positive control *ku+ , the zeros of ( ) 1ks t +

be ,k j ( 1,2, ,2 kj N

= A , 1,2, ,k m= A ; and1 2, ,k j k j

< for 1 21 2 kj j N < ) which represent

the switching times corresponding to negative control *ku . Altogether ,k j

+ 's and ,k j 's

represent the switching times which uniquely determine *ku as follows:

*,2 1 ,2

1

,2 1 ,21

1( ) { [sgn( ) sgn( )]

2

[sgn( ) sgn( )]}.

k

k

N

k k j k jj

N

k j k jj

u t t t

t t

+

+ +

=

=

=

Let the switch vector for the k th component of the control vector be

( ) ( )k k kN N N

+ =

where ,1 ,2k

k

Nk k N

+

++ + =

A and ,1 ,2

k

k

Nk k N

=

A . Let

2 2k k k

N N N+ = + . Then kN is the switching vector ofk

N dimensions.

Let the vector of switch functions for the control variable ku be defined as

1 2 2 1 2 2k k k k k

k k k k

N N N N N

N N N N + + + + +

= A A where 1 ,( 1) ( ( ) 1)

kN jk k k jj e s

+= +

( 1,2, ,2 kj N+= A ), and ,2 ( 1) ( ( ) 1)

k

k

N jk k k jj N

e s +

+= ( 1,2, ,2 kj N

= A ).

The gradient that can be used to update the switching vector kN can be given by

k

Nk

NJ

=

The optimal switching vector can be obtained iteratively by using a gradient-based method.

, 1 , ,k k kN i N i N k iK + = +

where ,k iK is a properly chosen k kN N -dimensional diagonal matrix with non-negative

entries for the i th iteration of the iterative optimization process; and ,kN i represents the

i th iteration of the switching vector kN .

When the optimal switching vectors are determined upon convergence, the optimal controltrajectories and the optimal state trajectories are computed. This process will be repeated for

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all selected nominal cases until all needed off-line optimal control and state trajectories areobtained. These trajectories will be used in training the fuel-optimal control oriented neuralnetworks.

3.3 Numerical solution scheme to the quadratic optimal control problemThe Hamiltonian function can be defined as

1( , , ) ( ( ) ( )) ( )

2e eH x u t x Qx u u R u u a Cp Bu

= + + + +

The state equation is given by

Hx a Cp Bu

= = + +

$

The costate equation can be given by

( )a Cp BuHQx

x x

+ +

= = +

$

The stationarity equation gives

( )0 ( )e

a Cp BuHR u u

u u

+ +

= = +

u can be solved out as

1 eu R B u= +

The Hamiltonian system becomes

1

1

( ) ( ) ( )( )

( ( ) ( ) ( )( ))

e

e

x a x C x p B x R B u

a x C x p B x R B uQx

x

= + + + + + +

= +

$

$

Furthermore, the boundary condition can be given by

( ) ( )( ( ) ( ))f f f ft S t x t r t =

Notice that for the Hamiltonian system which is composed of the state and costateequations, the initial condition is given for the state equation, and the constraints on thecostate variables at the final time for the costate equation.

It is observed that the Hamiltonian system is a set of nonlinear ordinary differential

equations in ( )x t and ( )t which develop forward and back in time, respectively. Generally,

it is not possible to obtain the analytic closed-form solution to such a two-point boundary-

value problem (TPBVP). Numerical methods have to be employed to solve for the

Hamiltonian system. One simple method, called shooting method may be used. There are

other methods like the shooting to a fixed point method, and relaxation methods, etc.

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The idea for the shooting method is as follows:1. First make a guess for the initial values for the costate.2. Integrate the Hamiltonian system forward.3. Evaluate the mismatch on the final constraints.

4. Find the sensitivity Jacobian for the final state and costate with respect to the initialcostate value.

5. Using the Newton-Raphson method to determine the change on the initial costatevalue.

6. Repeat the loop of steps 2 through 5 until the mismatch is close enough to zero.

4. Unified hierarchical neural control design framework

Keeping in mind that the discussions and analyses made in Section 3 are focused on the

system with a fixed parameter vector, which is the control problem ( 0P ). To address the

original control problem ( P ), the parameter vector space is tessellated into a number of sub-

regions. Each sub-region is identified with a set of vertexes. For each of the vertexes, a

different control problem ( 0P ) is formed. The family of control problems ( 0P ) are combined

together to represent an approximately accurate characterization of the dynamic system

behaviours exhibited by the nonlinear systems in the control problem ( P ). This is an

important step toward the hierarchical neural control design framework that is proposed to

address the optimal control of uncertain nonlinear systems.

4.1 Three-layer approach

While the control problem ( P ) is approximately equivalent to the family of control

problems ( 0P ), the solutions to the respective control problems ( 0P ) must be properly

coordinated in order to provide a consistent solution to the original control problem ( P ).

The requirement of consistent coordination of individual solutions may be mapped to the

hierarchical neural network control design framework proposed in this Chapter that

features the following: For a fixed parameter vector, the control solution characterized by a set of optimal state

and control trajectories shall be approximated by a neural network, which may becalled a nominal neural network for this nominal case. For each nominal case, anominal neural network is needed. All the nominal neural network controllersconstitute the nominal layer of neural network controllers.

For each sub-region, regional coordinating neural network controllers are needed to

coordinate the responses from individual nominal neural network controllers for thesub-region. All the regional coordinating neural network controllers constitute theregional layer of neural network controllers.

For an unknown parameter vector, global coordinating neural network controllers areneeded to coordinate the responses from regional coordinating neural networkcontrollers. All the global coordinating neural network controllers constitute the globallayer of neural networks controllers.

The proposed hierarchical neural network control design framework is a systematicextension and a comprehensive enhancement of the previous endeavours (Chen, 1998; Chen& Mohler & Chen, 2000).

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4.2 Nominal layer

Even though the hierarchical neural network control design methodology is unified and

generic, the design of the three layers of neural networks, especially the nominal layer of

neural networks may consider the uniqueness of the problems under study. For the time

optimal control problems, the role of the nominal layer of neural networks is to identify theswitching manifolds that relate to the bang-bang control. For the fuel optimal problems, the

role of the nominal layer of neural networks is to identify the switching manifolds that relateto the bang-off-bang control. For the quadratic optimal control problems, the role of the

nominal layer of neural networks is to approximate the optimal control based on the state

variables.

Fig. 1. Nominal neural network for time optimal control

Consequently a nominal neural network for the time optimal control takes the form of a

conventional neural network with continuous activation functions cascaded by a two-levelstair case function which itself may viewed as a discrete neural network itself, as shown in

Fig. 1. For the fuel optimal control, a nominal neural network takes the form of a

conventional neural network with continuous activation functions cascaded by a three-levelstair case function, as shown in Fig. 2.

Fig. 2. Nominal neural network for fuel optimal control

For the quadratic optimal control, no switching manifolds are involved. A conventionalneural network with continuous activation functions is sufficient for a nominal case, asshown in Fig. 3.

Conventional NN

Conventional NN

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Fig. 3. Nominal neural network for quadratic optimal control

4.3 Overall architecture

The overall architecture of the multi-layered hierarchical neural network control framework,as shown in Fig. 4, include three layers: the nominal layer, the regional layer, and the global

layer. These three layers play different roles and yet work together to attempt to achieve

desired control performance.At the nominal layer, the nominal neural networks are responsible to compute the near

optimal control signals for a given parameter vector. The post-processing function block is

necessary for both time optimal control problem and fuel optimal control problems while

indeed it may not be needed for the quadratic optimal control problems. For time optimal

control problems, the post-processing function is a sign function as shown in Fig. 2. For the

fuel optimal control problems, the post-processing is a slightly more complicated stair-case

function as shown in Fig. 3.

At the regional layer, the regional neural networks are responsible to compute the desired

weighting factors that are in turn used to modulate the control signals computed by the

nominal neural networks to produce near optimal control signals for an unknown

parameter vector situated at the know sub-region of the parameter vector space. The post-

processing function block is necessary for all the three types of control problems studied in

this Chapter. It is basically a normalization process of the weighting factors produced by the

regional neural networks for a sub-region that is enabled by the global neural networks.

At the global layer, the global neural networks are responsible to compute the possibilities

of the unknown parameter vector being located within sub-regions. The post-processing

function block is necessary for all the three types of control problems studied in this

Chapter. It is a winner-take-all logic applied to all the output data of the global neural

networks. Consequently, only one sub-regional will be enabled, and all the other sub-

regions will be disabled. The output data of the post-processing function block is used to

turn on only one of the sub-regions for the regional layer.To make use of the multi-layered hierarchical neural network control design framework, itis clear that the several key factors such as the number of the neural networks for each layer,

the size of each neural network, and desired training patterns, are important. This all has to

do with the determination of the nominal cases. A nominal case designates a group ofsystem conditions that reflect one of the typical system behaviors. In the context of control of

a dynamic system with uncertain parameters, which is the focus of this Chapter, a nominal

case may be designated as corresponding to the vertexes of the sub-regions when theparameter vector space is tessellated into a number of non-overlapping sub-regions down to

a level of desired granularity.

Conventional NN

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Fig. 4. Multi-layered hierarch neural network architecture

Post-Processingof RegionalLayer NNs

Post-Processing

of NominalLayer NNs

Regional Layer Neural Networks

Nominal Layer Neural Networks

Multiplication

Input

Post-Processingof GlobalLayer NNs

Global Layer Neural Networks

Nominal

Layer

RegionalLayer

GlobalLayer

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Once the nominal cases are identified, the numbers of neural networks for the nominal layer,the regional layer and the global layer can be determined accordingly. Each nominal neuralnetwork corresponds to a nominal case identified. Each regional neural network correspondsto a nominal neural network. Each global neural network corresponds to a sub-region.

With the numbers of neural networks for all the three layers in the hierarchy determined,the size of each neural network is dependent upon the data collected for each nominal case.As shown in the last Section, the optimal state trajectories and the optimal control

trajectories for each of the control problems ( 0P ) can be obtained through use of the STVM

approach for time optimal control and for fuel optimal control or the shooting method forthe quadratic optimal control. For each of the nominal cases, the optimal state trajectoriesand optimal control trajectories may be properly utilized to form the needed trainingpatterns.

4.4 Design procedure

Below is the design procedure for multi-layered hierarchical neural networks: Identify the nominal cases. The parameter vector space may be tessellated into a

number of non-overlapping sub-regions. The granualarity of the tessellation process isdetermined by how sensitive the system dynamic behaviors are to the changes of theparameters. Each vertext of the sub-regions identifies a nominal case. For each nominalcase, the optimal control problem may be solved numerically and the nuermicalsolution may be obtained.

Determine the size of the nominal layer, the regional layer and the global layer of thehierarchy.

Determine the size of the neural networks for each layer in the hierarchy. Train the nominal neural networks. The numerically obtained optimal state and control

trajectories are acquired for each nominal case. The training data pattern for thenominal neural networks is composed of the state vector as input and the control signalas the output. In other words, the nominal layer is to establish and approximate a statefeedback control. Finish training when the training performance is satisfactory. Repeatthis nominal layer training process for all the nominal neural networks.

Training the regional neural networks. The input data to the nominal neural networksis also part of the input data to the regional neural networks. In addition, for a specificregional neural network, the ideal output data of the corresponding nominal neuralnetwork is also part of its input data. The ideal output data of the regional neuralnetwork can be determined as follows:

If the data presented to a given regional neural network reflects a nominal case thatcorresponds to the vertex that this regional neural network is to be trained for, thenassign 1 or else 0.

Training the global neural networks. The input data to the nominal neural networks isalso part of the input data to the global neural networks. In addition, for a specificglobal neural network, the ideal output data of the corresponding nominal neuralnetwork is also part of its input data. The ideal output data of the global neural networkcan be determined as follows:

If the data presented to a given global neural network reflects a nominal case thatcorresponds to the sub-region that this global neural network is to be trained for,then assign 1 or else 0.

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5. Theoretical justification

This Section provides theoretical support for the adoption of the hierarchical neuralnetworks.

As shown in (Chen, Yang & Moher, 2006), the desired prediction or control can be achievedby a properly designed hierarchical neural network.Proposition 1 (Chen, Yang & Mohler, 2006): Suppose that an ideal system controller can be

characterized by function vectors ui andl

i ( 1 l ui n n = ) which are continuous

mappings from a compact support xnR to yn

R , such that a continuous function vector

falso defined on can be expressed as , ,1( ) ( ) ( )ln u l

j i j i jix f x f x

== on the point-wise basis

( x ; and , ( )ui j x and , ( )

li j x are the jth component of

uif and

li ). Then there exists a

hierarchical neural network, used to approximate the ideal system controller or system

identifier, that includes lower level neural networks linn 's and upper level neural networks

uinn ( 1 l ui n n = ) such that for any 0j > , , ,1sup | |

ln l ux j i j i j ji

f nn nn =


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Define 1 2( ) ( ) ( )x t x t x t = , and p p p = . Then we have

0

01

( ) { ( ) ( ) ( )] ( ) }

( ( ))

t

T T Tt

t

t

x t a x s B x s u s C x s p ds

C x s pds

= + + +

If the both sides of the above equation takes an appropriate norm and the triangle inequalityis applied, the following is obtained:

0

01

|| ( )|||| { ( ) ( ) ( )] ( ) } ||

|| ( ( )) ||

t

T T Tt

t

t

x t a x s B x s u s C x s p ds

C x s p ds

+ + +

Note that 1|| ( ( ) ||C x s p

can be made uniformly bounded by as long as the estimate ofp is made sufficiently close to p (which can be controlled by the granularity of tessellation),

and p is bounded; | ( )| 1u t ; || || sup ( )T x Ta a x= < , || || sup ( )T x TB B x= < and

|| || sup ( )T x TC C x= < .

It follows that

00|| ( )|| ( ) (|| || || || || |||| ||) ( )

t

T T T tx t t t a B C p x s ds + + +

Define a constant 0 (|| || || || || |||| ||)T T TK a B C p= + + . Applying the Gronwall-Bellman

Inequality to the above inequality yields

00 0 0 0

20

0 0 0 0

|| ( )|| ( ) ( )exp{ }

( )( ) exp( ( ))

2

t t

t sx t t t K s t K d ds

t tt t K K t t K

+

+

where0

0 0 0 0

( )( )(1 exp( ( )))

2

t tK t t K K t t

= + , and K< .

This completes the proof.

6. Simulation

Consider the single-machine infinity-bus (SMIB) model with a thyristor-controlled series-capacitor (TCSC) installed on the transmission line (Chen, 1998) as shown in Fig. 5, whichmay be mathematically described as follows:

0

( 1)

1( ( 1) sin )

(1 )

b

tm

d e

VVP P D

M X s X

= +

$

$

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where is rotor angle (rad), rotor speed (p.u.), 2 60b = synchronous speed as base

(rad/sec), 0.3665mP = is mechanical power input (p.u.), 0P is unknown fixed load (p.u.),

2.0D = damping factor, 3.5M= system inertia referenced to the base power, 1.0tV =

terminal bus voltage (p.u.), 0.99V = infinite bus voltage (p.u.), 2.0dX = transientreactance of the generator (p.u.), 0.35eX = transmission reactance (p.u.),

min max[ , ] [0.2,0.75]s s s = series compensation degree of the TCSC, and ( ,1)e is system

equilibrium with the series compensation degree fixed at 0.4es = .

The goal is to stabilize the system in the near optimal time control fashion with an

unknown load 0P ranging 0 and 10% of mP . Two nominal cases are identified. The

nominal neural networks have 15 and 30 neurons in the first and second hidden layer

with log-sigmoid and tan-sigmoid activation functions for these two hidden layers,

respectively. The input data to regional neural networks is the rotor angle, its two

previous values, the control and its previous value, and the outputs are the weighting

factors. The regional neural networks have 15 and 30 neurons in the first and second

hidden layer with log-sigmoid and tan-sigmoid activation functions for these two hidden

layers, respectively. The global neural networks are really not necessary in this simple

case of parameter uncertainty.Once the nominal and regional neural networks are trained, they are used to control the

system after a severe short-circuit fault and with an unknown load (5% of mP ). The resulting

trajectory is shown in Fig. 6. It is observed that the hierarchical neural controller stabilizesthe system in a near optimal control manner.

Fig. 5. The SMIB system with TCSC

SynchronousMachine

TransmissionLine with TCSC

InfiniteBus

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Fig. 6. Control performance of hierarchical neural controller. Solid - neural control; dashed -optimal control.

7. Conclusion

Even with remarkable progress witnessed in the adaptive control techniques for the

nonlinear system control over the past decade, the general challenge with adaptive control

of nonlinear systems has never become less formidable, not to mention the adaptive controlof nonlinear systems while optimizing a pre-designated control performance index and

respecting restrictions on control signals. Neural networks have been introduced to tackle

the adaptive control of nonlinear systems, where there are system uncertainties in

parameters, unmodeled nonlinear system dynamics, and in many cases the parameters maybe time varying. It is the main focus of this Chapter to establish a framework in which

general nonlinear systems will be targeted and near optimal, adaptive control of uncertain,time-varying, nonlinear systems is studied. The study begins with a generic presentation of

the solution scheme for fixed-parameter nonlinear systems. The optimal control solution is

presented for the purpose of minimum time control and minimum fuel control, respectively.

The parameter space is tessellated into a set of convex sub-regions. The set of parameter

vectors corresponding to the vertexes of those convex sub-regions are obtained.Accordingly, a set of optimal control problems are solved. The resulting control trajectories

and state or output trajectories are employed to train a set of properly designed neuralnetworks to establish a relationship that would otherwise be unavailable for the sake of near

optimal controller design. In addition, techniques are developed and applied to deal with

the time varying property of uncertain parameters of the nonlinear systems. All these pieces

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come together in an organized and cooperative manner under the unified intelligent controldesign framework to meet the Chapters ultimate goal of constructing intelligent controllers

for uncertain, nonlinear systems.

8. Acknowledgment

The authors are grateful to the Editor and the anonymous reviewers for their constructivecomments.

9. References

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Network, Proceedings of 1997 American Control Conference, pp. 1086-1090, ISBN 0-

7803-3832-4, Albuquerque, New Mexico, USA, June 4-6, 1997.Chen, D. & Mohler, R. (2000). Theoretical Aspects on Synthesis of Hierarchical Neural

Controllers for Power Systems, Proceedings of 2000 American Control Conference, pp.

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Chen, D. & Mohler, R. (2003). Neural-Network-based Loading Modeling and Its Use in

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July 2005.

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Sanner, R. & Slotine, J. (1992). Gaussian Networks for Direct Adaptive Control. IEEE

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Zakrzewski, R. R., Mohler, R. R., & Kolodziej, W. J. (1994). Hierarchical Intelligent Controlwith Flexible AC Transmission System Application. IFAC Journal of Control

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Recent Advances in Robust Control - Novel Approaches and

Design Methods

Edited by Dr. Andreas Mueller

ISBN 978-953-307-339-2

Hard cover, 462 pages

Publisher InTech

Published online 07, November, 2011

Published in print edition November, 2011

InTech Europe

University Campus STeP Ri

Slavka Krautzeka 83/A

51000 Rijeka, Croatia

Phone: +385 (51) 770 447

Fax: +385 (51) 686 166

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InTech China

Unit 405, Office Block, Hotel Equatorial Shanghai

No.65, Yan An Road (West), Shanghai, 200040, China

Phone: +86-21-62489820

Fax: +86-21-62489821

Robust control has been a topic of active research in the last three decades culminating in H_2/H_\infty and

\mu design methods followed by research on parametric robustness, initially motivated by Kharitonov's

theorem, the extension to non-linear time delay systems, and other more recent methods. The two volumes of

Recent Advances in Robust Control give a selective overview of recent theoretical developments and present

selected application examples. The volumes comprise 39 contributions covering various theoretical aspects as

well as different application areas. The first volume covers selected problems in the theory of robust control

and its application to robotic and electromechanical systems. The second volume is dedicated to special topics

in robust control and problem specific solutions. Recent Advances in Robust Control will be a valuable

reference for those interested in the recent theoretical advances and for researchers working in the broad field

of robotics and mechatronics.

How to reference

In order to correctly reference this scholarly work, feel free to copy and paste the following:

Dingguo Chen, Lu Wang, Jiaben Yang and Ronald R. Mohler (2011). Neural Control Toward a Unified

Intelligent Control Design Framework for Nonlinear Systems, Recent Advances in Robust Control - Novel

Approaches and Design Methods, Dr. Andreas Mueller (Ed.), ISBN: 978-953-307-339-2, InTech, Available

from: http://www.intechopen.com/books/recent-advances-in-robust-control-novel-approaches-and-design-

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